1. The Unwavering Recommendation of Telegram Desktop:

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As of this very second, the Telegram Desktop application stands as the most reliable and feature-complete tool for exporting your Telegram history. By installing it on a computer (Windows, macOS, or Linux), logging in, and utilizing the "Export chat history" function for specific conversations or the "Export Telegram data" option for broader exports, users can obtain their data in formats like HTML. This exported data can then be transferred to an Android device if so desired.
2. The Persistent Limitation of Saving Individual Content:

The inherent Android app functionalities for saving new zealand telegram phone number list individual messages ("Forward to Saved Messages") and manually saving media files remain the direct, albeit highly limited, ways to preserve specific elements of your Telegram history directly on your phone. This approach is not a viable solution for exporting entire conversations or comprehensive data sets.